*Coordinate Transformations with t*

*x* = space coordinate, *t* = time coordinate, *v* = velocity, *u* = pace

Galilean transformation

speed: *x´* = *x* – *vt* and *t**´* = *t*, pace: *x´* = *x* – *t/u* and *t**´* = *t*.

Dual Galilean transformation

speed: *t´* = *t* – *x/v* and *x**´* = *x*, pace: *t´* = *t* – *ux* and *x**´* = *x*.

Light: *c* := speed of light, *c´* := pace of light.

speed: *x* = *ct* or* x/c = t* and *x´* = *ct´*, or *x´/c* = *t´*,

pace: *c´x* = *t* or *x* = *t/c´* and *c´r′ *= *t´* or *x´ *= *t´/c´*.

Lorentz transformation

speed: *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2} with *x**´* = *γ* (*x − vt*) and *t´* = *γ* (*t* – *xv/c²*),

pace: *γ* = (1 – *c´*^{2}/*u*^{2})^{–1/2} with *x´* = *γ* (*x* – *t/u*) and *t**´* = *γ* (*t* – *xc´²/u*),

which applies only if |*v*| < |*c*| or |*u*| > |*c´*|.

Dual Lorentz transformation

speed: *λ* = (1 − *c*^{2}/*v*^{2})^{–1/2} with *t´* = *λ* (*t − x/v*) and *x´* = *λ* (*x* − *t* (*c*^{2}*/v*)),

pace: *λ* = (1 – *u*^{2}/*c´*^{2})^{–1/2} with *t´* = *λ* (*t* – *ux*) and *x´ *= *λ* (*x* – *t* (*u/c´²*)),

which applies only if |*v*| > |*c*| or |*u*| < |*c´*|.

Galileo-Lorentz limit

speed: *c*_{1} → ∞: *γ*_{1} → 1, *t**´* → *t*, *c*_{2} → *c*/2: *γ*_{2} → (1 – 4*v*²/*c*²)^{−1/2} and *t**´* → *γ*_{2} (*t* – 4*x* (*v*/*c*²)).

pace: *c´*_{1} → 0: *γ*_{1} → 1, *t**´* → *t*, *ç*_{2} → 2/*c´*: *γ*_{2} → (1 – (1/4) *c´*²/*u*²)^{−1/2} and *t*′ → *γ*_{2} (*t* – (*x*/4) (*c´*²/*u*)).

Dual Galileo-Lorentz limit

speed: *c*_{1} → 0: *λ*_{1} → 1, *x**´* → *x*, *c*_{2} → 2/c: *λ*_{2} → (1 – (1/4) *c*²/*v*²)^{−1/2} and *x**´* → *λ*_{2} (*x* – (*t*/4) (*c*²/*v*)).

pace: *c´*_{1} → ∞: *λ*_{1} → 1, *x**´* → *x*, *c´*_{2} → *c´*/2: *λ*_{2} → (1 – 4*u*²/*c´*²)^{−1/2} and *x**´* → *λ*_{2} (*x* – 4*t* (*u*/*c´*²)).

If |*v*| = |*c*|, then *x′ = x* and *t′ = t*.

*Coordinate Transformations with τ*

*x* = space coordinate, *t* = time coordinate, *v* = velocity, *u* = pace

*c* := speed of light, *c´* := pace of light.

*τ* := *ct τ* := *t*/*c´*

Galilean transformation

speed: *x´* = *x* – *τ* (*v*/*c*) and *τ**´* = *τ*, pace: *x´* = *x* – *τ (c´*/*u*) and *τ**´* = *τ*.

Dual Galilean transformation

speed: *τ´* = *τ* – *x* (*c*/*v*) and *x**´* = *x*, pace: *τ´* = *τ* – *x* (*u*/*c´*) and *x**´* = *x*.

Light: *x* = *τ* and *x´* = *τ´*

Lorentz transformation

speed: *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2} with *x´* = *γ* (*x − **τ* (*v*/*c*)) and *τ´* = *γ* (*τ* – *x* (*v*/*c*)),

pace: *γ* = (1 – *c´*^{2}/*u*^{2})^{–1/2} with *x´* = *γ* (*x* – *τ* (*c´*/*u*)) and *τ´* = *γ* (*τ* – *x* (*c´*/*u*)),

which applies only if |*v*| < |*c*| or |*u*| > |*c´*|.

Dual Lorentz transformation

speed: *λ* = (1 − *c*^{2}/*v*^{2})^{–1/2} with *τ′* = *λ* (*τ – x* (*c*/*v*)) and *x´* = *λ* (*x* − *τ* (*c*/*v*)),

pace: *λ* = (1 – *u*^{2}/*c´*^{2})^{–1/2} with *τ′* = *λ* (*τ* – *x* (*u*/*c´*)) and *x´ *= *λ* (*x* – *τ* (*u*/*c´*)),

which applies only if |*v*| > |*c*| or |*u*| < |*c´*|.

Galileo-Lorentz limit

speed: *c*_{1} → ∞: *γ*_{1} → 1, *τ**´* → *τ*, *c*_{2} → *c*/2: *γ*_{2} → (1 – 4*v*²/*c*²)^{−1/2} and *τ*′ → *γ*_{2} (*τ* – 4*x* (*v*/*c*)).

pace: *ç*_{1} → 0: *γ*_{1} → 1, *τ**´* → *τ*, *ç*_{2} → 2/*c´*: *γ*_{2} → (1 – (1/4) *c´*²/*u*²)^{−1/2} and *τ*′ → *γ*_{2} (*τ* – (*x*/4) (*c´*/*u*)).

Dual Galileo-Lorentz limit

speed: *c*_{1} → 0: *λ*_{1} → 1, *x**´* → *x*, *c*_{2} → 2/c: *λ*_{2} → (1 – (1/4) *c*²/*v*²)^{−1/2} and *x**´* → *λ*_{2} (*x* – (*τ*/4) (*c*/*v*)).

pace: *ç*_{1} → ∞: *λ*_{1} → 1, *x**´* → *x*, *ç*_{2} → *c´*/2: *λ*_{2} → (1 – 4*u*²/*c´*²)^{−1/2} and *x**´* → *λ*_{2} (*x* – 4*τ* (*u*/*c´*)).

If |*v*| = |*c*|, then *x′ = x* and *τ′ = τ*.

*Coordinate Transformations with c := 1*

*x* = space coordinate, *t* = time coordinate, *v* = velocity, *u* = pace

Galilean transformation

speed: *x´* = *x* – *vt* and *t*´ = *t*, pace: *x´* = *x* – *t/u* and *t*´ = *t*.

Dual Galilean transformation

speed: *t´* = *t* – *x/v* and *x*´ = *x*, pace: *t´* = *t* – *ux* and *x*´ = *x*.

Light: 1 = *c* := speed of light, 1 = *c´* := pace of light.

speed: *x* = *t* and *x*´ = *t´*

pace: *x* = *t* and *x´ *= *t´*.

Lorentz transformation

speed: *γ* = (1 – *v*^{2})^{–1/2} with *x´* = *γ* (*x − vt*) and *t´* = *γ* (*t* – *xv*),

pace: *γ* = (1 – 1/*u*^{2})^{–1/2} with *x´* = *γ* (*x* – *t/u*) and *t´* = *γ* (*t* – *x/u*),

which applies only if |*v*| < 1 or |*u*| > 1.

Dual Lorentz transformation

speed: *λ* = (1 − 1/*v*^{2})^{–1/2} with *t´* = *λ* (*t − x/v*) and *x´* = *λ* (*x* − *t* (1*/v*)),

pace: *λ* = (1 – *u*^{2})^{–1/2} with *t′* = *λ* (*t* – *ux*) and *x´ *= *λ* (*x* – *tu*),

which applies only if |*v*| > 1 or |*u*| < 1.

If |*v*| = 1, then *x´ = x* and *t´ = t*.